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Differential Equations : Their Solution Using Symmetries - Hans Stephani

Differential Equations

Their Solution Using Symmetries

Paperback

Published: 8th October 1990
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In many branches of physics, mathematics, and engineering, solving a problem means solving a set of ordinary or partial differential equations. Nearly all methods of constructing closed form solutions rely on symmetries. The emphasis in this text is on how to find and use the symmetries; this is supported by many examples and more than 100 exercises. This book will form an introduction accessible to beginning graduate students in physics, applied mathematics, and engineering. Advanced graduate students and researchers in these disciplines will find the book a valuable reference.

"...as an account of classical general relativity, this well produced and excellently translated book has many virtues." Physics Bulletin "A nice introduction to the theory and practice of finding and using symmetries to solve differential equations." American Mathematical Monthly "...Stephani's book does a good job of motivating the study of Lie group methods for differential equations from an elementary standpoint." SIAM Reviews "The author, who has an easy-to-read practical style, consistently keeps the emphasis on applications. Thus, most physicists will be able to get useful information about ordinary differential equations (ODE's) and partial differential equations (PDE's), without being bogged down in cumbersome mathematical formalism...well worth reading if one is at all interested in sophisticated and powerful symmetry techniques for handling differential equations and if one wishes to have the most straightforward approach to the topic." D. E. Vincent, Physics in Canada "...Stephani...has built a book that tries to guide its readers toward a sure knowledge of this very important tool for finding solutions of (nonlinear) differential equations. In the early sections, the derivations presented are the most clear and detailed ones that this writer has ever seen...Students new to this area will find reading or studying the current book an altogether enjoyable occupation, without any of the intimidation that sometimes is caused by similar books." J.D. Finley, Foundations of Physics

Prefacep. xi
Introductionp. 1
Ordinary differential equations
Point transformations and their generatorsp. 5
One-parameter groups of point transformations and their infinitesimal generatorsp. 5
Transformation laws and normal forms of generatorsp. 9
Extensions of transformations and their generatorsp. 11
Multiple-parameter groups of transformations and their generatorsp. 14
Exercisesp. 16
Lie point symmetries of ordinary differential equations: the basic definitions and propertiesp. 17
The definition of a symmetry: first formulationp. 17
Ordinary differential equations and linear partial differential equations of first orderp. 20
The definition of a symmetry: second formulationp. 22
Summaryp. 25
Exercisesp. 25
How to find the Lie point symmetries of an ordinary differential equationp. 26
Remarks on the general procedurep. 26
The atypical case: first order differential equationsp. 27
Second order differential equationsp. 28
Higher order differential equations. The general nth order linear equationp. 33
Exercisesp. 36
How to use Lie point symmetries: differential equations with one symmetryp. 37
First order differential equationsp. 37
Higher order differential equationsp. 39
Exercisesp. 45
Some basic properties of Lie algebrasp. 46
The generators of multiple-parameter groups and their Lie algebrasp. 46
Examples of Lie algebrasp. 49
Subgroups and subalgebrasp. 51
Realizations of Lie algebras. Invariants and differential invariantsp. 53
Nth order differential equations with multiple-parameter symmetry groups: an outlookp. 57
Exercisesp. 58
How to use Lie point symmetries: second order differential equations admitting a G[subscript 2]p. 59
A classification of the possible subcases, and ways one might proceedp. 59
The first integration strategy: normal forms of generators in the space of variablesp. 62
The second integration strategy: normal forms of generators in the space of first integralsp. 66
Summary: Recipe for the integration of second order differential equations admitting a group G[subscript 2]p. 69
Examplesp. 70
Exercisesp. 74
Second order differential equations admitting more than two Lie point symmetriesp. 75
The problem: groups that do not contain a G[subscript 2]p. 75
How to solve differential equations that admit a G[subscript 3] IXp. 76
Examplep. 78
Exercisesp. 79
Higher order differential equations admitting more than one Lie point symmetryp. 80
The problem: some general remarksp. 80
First integration strategy: normal forms of generators in the space(s) of variablesp. 81
Second integration strategy: normal forms of generators in the space of first integrals. Lie's theoremp. 83
Third integration strategy: differential invariantsp. 87
Examplesp. 88
Exercisesp. 92
Systems of second order differential equationsp. 93
The corresponding linear partial differential equation of first order and the symmetry conditionsp. 93
Example: the Kepler problemp. 96
Systems possessing a Lagrangian: symmetries and conservation lawsp. 97
Exercisesp. 100
Symmetries more general than Lie point symmetriesp. 101
Why generalize point transformations and symmetries?p. 101
How to generalize point transformations and symmetriesp. 103
Contact transformationsp. 105
How to find and use contact symmetries of an ordinary differential equationp. 107
Exercisesp. 109
Dynamical symmetries: the basic definitions and propertiesp. 110
What is a dynamical symmetry?p. 110
Examples of dynamical symmetriesp. 112
The structure of the set of dynamical symmetriesp. 114
Exercisesp. 116
How to find and use dynamical symmetries for systems possessing a Lagrangianp. 117
Dynamical symmetries and conservation lawsp. 117
Example: L = (x[superscript 2] + y[superscript 2])/2 - a(2y[superscript 3] + x[superscript 2]y), a [characters not producible] 0p. 119
Example: the Kepler problemp. 121
Example: geodesics of a Riemannian space - Killing vectors and Killing tensorsp. 123
Exercisesp. 127
Systems of first order differential equations with a fundamental system of solutionsp. 128
The problemp. 128
The answerp. 129
Examplesp. 131
Systems with a fundamental system of solutions and linear systemsp. 134
Exercisesp. 137
Partial differential equations
Lie point transformations and symmetriesp. 141
Introductionp. 141
Point transformations and their generatorsp. 142
The definition of a symmetryp. 145
Exercisesp. 147
How to determine the point symmetries of partial differential equationsp. 148
First order differential equationsp. 148
Second order differential equationsp. 154
Exercisesp. 160
How to use Lie point symmetries of partial differential equations I: generating solutions by symmetry transformationsp. 161
The structure of the set of symmetry generatorsp. 161
What can symmetry transformations be expected to achieve?p. 163
Generating solutions by finite symmetry transformationsp. 164
Generating solutions (of linear differential equations) by applying the generatorsp. 167
Exercisesp. 169
How to use Lie point symmetries of partial differential equations II: similarity variables and reduction of the number of variablesp. 170
The problemp. 170
Similarity variables and how to find themp. 171
Examplesp. 174
Conditional symmetriesp. 179
Exercisesp. 182
How to use Lie point symmetries of partial differential equations III: multiple reduction of variables and differential invariantsp. 184
Multiple reduction of variables step by stepp. 184
Multiple reduction of variables by using invariantsp. 189
Some remarks on group-invariant solutions and their classificationp. 191
Exercisesp. 192
Symmetries and the separability of partial differential equationsp. 193
The problemp. 193
Some remarks on the usual separations of the wave equationp. 194
Hamilton's canonical equations and first integrals in involutionp. 196
Quadratic first integrals in involution and the separability of the Hamilton-Jacobi equation and the wave equationp. 199
Exercisesp. 201
Contact transformations and contact symmetries of partial differential equations, and how to use themp. 202
The general contact transformation and its infinitesimal generatorp. 202
Contact symmetries of partial differential equations and how to find themp. 204
Remarks on how to use contact symmetries for reduction of variablesp. 206
Exercisesp. 208
Differential equations and symmetries in the language of formsp. 209
Vectors and formsp. 209
Exterior derivatives and Lie derivativesp. 212
Differential equations in the language of formsp. 213
Symmetries of differential equations in the language of formsp. 215
Exercisesp. 219
Lie-Backlund transformationsp. 220
Why study more general transformations and symmetries?p. 220
Finite order generalizations do not existp. 223
Lie-Backlund transformations and their infinitesimal generatorsp. 225
Examples of Lie-Backlund transformationsp. 227
Lie-Backlund versus Backlund transformationsp. 229
Exercisesp. 231
Lie-Backlund symmetries and how to find themp. 232
The basic definitionsp. 232
Remarks on the structure of the set of Lie-Backlund symmetriesp. 233
How to find Lie-Backlund symmetries: some general remarksp. 236
Examples of Lie-Backlund symmetriesp. 237
Recursion operatorsp. 242
Exercisesp. 245
How to use Lie-Backlund symmetriesp. 246
Generating solutions by finite symmetry transformationsp. 246
Similarity solutions for Lie-Backlund symmetriesp. 248
Lie-Backlund symmetries and conservation lawsp. 250
Lie-Backlund symmetries and generation methodsp. 251
Exercisesp. 252
A short guide to the literaturep. 253
Solutions to some of the more difficult exercisesp. 255
Indexp. 259
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521366892
ISBN-10: 0521366895
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 276
Published: 8th October 1990
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 24.13 x 15.24  x 1.91
Weight (kg): 0.45